Change Viewport's Orbiting Axis While Sculpting
A workaround would be to parent a camera to an empty, as follows:



Then, in the viewport snapping settings, enable face snapping and check Align Rotation to Target.



After doing this, dragging the empty around will snap the camera perpendicular to whichever face your cursor is on.



Setting the transformation orientation to Local will allow you to transform the empty relative to its own coordinate system.



Additionally, you can enable the visibility of gizmos to be able to easily rotate the empty: